We call a number a mountain number if its middle digit is larger than any other digit.  For example, 284 is a mountain number.  How many 3-digit mountain numbers are there?
We will break this into three cases.

Case 1: numbers of the form $xyx$ ($x \ne 0$).

Any pair of nonzero digits has a corresponding palindrome ($xyx$) mountain number, so the number of these is $\binom{9}{2} = 36$.

Case 2: numbers of the form $xyz$ ($z \ne 0, x \ne z$).

Any group of three nonzero digits ($y > x > z > 0$) has two corresponding mountain numbers ($xyz$ and $zyx$), so the number of these is $2 \times \binom{9}{3} = 168$.

Case 3: numbers of the form $xy0$ ($x \ne 0, y \ne 0$).

Any pair of nonzero digits has a corresponding mountain number in the form $xy0$, so there are $\binom{9}{2} = 36$ of these.

So the total number of mountain numbers is $36 + 168 + 36 = \boxed{240}$.